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The Cauchy problem and the integral operator

Equation ([*]) describes a continuous process of reflected wavefield continuation in the time-offset-midpoint domain. In order to find an integral-type operator that performs the one-step offset continuation, I consider the following initial-value (Cauchy) problem for equation ([*]):

Given a post-NMO constant-offset section at half-offset h1  
\left.P(t_n,h,y)\right\vert _{h=h_1}=P^{(0)}_1(t_n,y)\end{displaymath} (219)
and its first-order derivative with respect to offset  
\left.\partial P(t_n,h,y)\over \partial h\right\vert _{h=h_1}=P^{(1)}_1(t_n,y)\;,\end{displaymath} (220)
find the corresponding section P(0)(tn,y) at offset h.

Equation ([*]) belongs to the hyperbolic type, with the offset coordinate h being a ``time-like'' variable and the midpoint coordinate y and the time tn being ``space-like'' variables. The last condition ([*]) is required for the initial value problem to be well-posed Courant (1962). From a physical point of view, its role is to separate the two different wave-like processes embedded in equation ([*]), which are analogous to inward and outward wave propagation. We will associate the first process with continuation to a larger offset and the second one with continuation to a smaller offset. Though the offset derivatives of data are not measured in practice, they can be estimated from the data at neighboring offsets by a finite-difference approximation. Selecting a propagation branch explicitly, for example by considering the high-frequency asymptotics of the continuation operators, can allow us to eliminate the need for condition ([*]). In this section, I discuss the exact integral solution of the OC equation and analyze its asymptotics.

The integral solution of problem ([*]-[*]) for equation ([*]) is obtained in Appendix [*] with the help of the classic methods of mathematical physics. It takes the explicit form

P(t_n,h,y) & = &
\int\!\!\int P^{(0)}_1(t_1,y_1)\,G_0(t_1,h_1,y...
 ...\!\int P^{(1)}_1(t_1,y_1)\,G_1(t_1,h_1,y_1;t_n,h,y)\,dt_1\,dy_1\;,\end{eqnarray}
where the Green's functions G0 and G1 are expressed as
G_0(t_1,h_1,y_1;t_n,h,y) & = & \mbox{sign}(h-h_1)\,{H(t_n) \ove...
 ..._n \over t_1^2}\,\left\{
H(\Theta) \over 
\sqrt{\Theta}\right\}\;,\end{eqnarray} (222)
and the parameter $\Theta$ is  
\Theta(t_1,h_1,y_1;t_n,h,y) = 
\left(y_1-y\right)^2\;.\end{displaymath} (224)
H stands for the Heavyside step-function.

From equations ([*]) and ([*]) one can see that the impulse response of the offset continuation operator is discontinuous in the time-offset-midpoint space on a surface defined by the equality  
\Theta(t_1,h_1,y_1;t_n,h,y) = 0\;,\end{displaymath} (225)
which describes the ``wavefronts'' of the offset continuation process. In terms of the theory of characteristics Courant (1962), the surface $\Theta=0$ corresponds to the characteristic conoid formed by the bi-characteristics of equation ([*]) - time rays emerging from the point $\{t_n,h,y\}=\{t_1,h_1,y_1\}$. The common-offset slices of the characteristic conoid are shown in the left plot of Figure [*].

Figure 7
Constant-offset sections of the characteristic conoid - ``offset continuation fronts'' (left), and branches of the conoid used in the integral OC operator (right). The upper part of the plots (small times) corresponds to continuation to smaller offsets; the lower part (large times) corresponds to larger offsets.
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As a second-order differential equation of the hyperbolic type, equation ([*]) describes two different processes. The first process is ``forward'' continuation from smaller to larger offsets, the second one is ``reverse'' continuation in the opposite direction. These two processes are clearly separated in the high-frequency asymptotics of operator ([*]). To obtain the asymptotical representation, it is sufficient to note that ${1
 \over \sqrt{\pi}}\, {H(t) \over \sqrt{t}}$ is the impulse response of the causal half-order integration operator and that $H(t^2-a^2)
\over \sqrt{t^2-a^2}$ is asymptotically equivalent to $H(t-a) \over
{\sqrt{2a}\,\sqrt{t-a}}$ (t, a >0). Thus, the asymptotical form of the integral offset-continuation operator becomes

P^{(\pm)}(t_n,h,y) & = &
{\bf D}^{1/2}_{\pm\,t_n}\,\int w^{(\pm...
Here the signs ``+'' and ``-'' correspond to the type of continuation (the sign of h-h1), ${\bf D}^{1/2}_{\pm\,t_n}$ and ${\bf I}^{1/2}_{\pm\,t_n}$ stand for the operators of causal and anticausal half-order differentiation and integration applied with respect to the time variable tn, the summation paths $\theta^{(\pm)}(\xi;h_1,h,t_n)$ correspond to the two non-negative sections of the characteristic conoid ([*]) (Figure [*]):

{t_n \over h}\,\sqrt{{U \pm V} \over 2 }\;,\end{displaymath} (227)
where $U=h^2+h_1^2-\xi^2$, and $V=\sqrt{U^2-4\,h^2\,h_1^2}$; $\xi$ is the midpoint separation (the integration parameter), and $w^{(\pm)}_0$and $w^{(\pm)}_1$ are the following weighting functions:

w^{(\pm)}_0 & = & {1 \over \sqrt{2\,\pi}}\,
 ...rt{t_n}\, h_1} \over {\sqrt{V}\,\theta^{(\pm)}(\xi;h_1,h,t_n)}}\;.\end{eqnarray} (228)
Expression ([*]) for the summation path of the OC operator was obtained previously by Stovas and Fomel (1993, 1996) and Biondi and Chemingui (1994a,b). A somewhat different form of it is proposed by Bagaini and Spagnolini (1996). I describe the kinematic interpretation of formula ([*]) in Appendix [*].

In the high-frequency asymptotics, it is possible to replace the two terms in equation ([*]) with a single term Fomel (1996b). The single-term expression is
P^{(\pm)}(t_n,h,y) = 
{\bf D}^{1/2}_{\pm\,t_n}\,\int w^{(\pm...
P^{(0)}_1(\theta^{(\pm)}(\xi;h_1,h,t_n),y_1-\xi)\,d\xi\;,\end{displaymath} (230)
w^{(+)} & = & \sqrt{\theta^{(+)}(\xi;h_1,h,t_n) \over {2\,\pi}}...
 ...\over \sqrt{2\,\pi t_n}}\;
{{h_1^2-h^2 +\xi^2} \over {V^{3/2}}}\;.\end{eqnarray} (231)
A more general approach to true-amplitude asymptotic offset continuation is developed by Santos et al. (1997).

The limit of expression ([*]) for the output offset h approaching zero can be evaluated by L'Hospitale's rule. As one would expect, it coincides with the well-known expression for the summation path of the integral DMO operator Deregowski and Rocca (1981)  
\lim_{h \rightarrow 0} {{t_...
 ...rt{{U - V} \over 2 }}=
{{t_n\,h_1} \over \sqrt{h_1^2-\xi^2}}\;.\end{displaymath} (233)
I discuss the connection between offset continuation and DMO in the next section.

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